You last point is probably true in some way, but I do know that airlines always have one eye on how much a flight will cost them in terms of fuel.
A lower cost index, the more the airline is trying to save fuel, climb slower, slower cruising speed, and therefore presumably lower cruising altitude.
A higher cost index places more stress on the mechanical components of the aircraft - burns more fuel, climbs quicker and likely higher cruising speed.
Airlines keep BOTH eyes on the fuel cost 99% of the time. Cost indexes are worked out when the flight plan is generated by a computer that knows all variables and simply punches out a number. Overwhelmingly, the cost of fuel is the biggest influence. A higher cost index costs virtually nothing mechanically. The cost index will affect climb, cruise and descent speeds but doesn’t much influence the cruise level. The FMC obviously knows the weight of the aircraft and will advise the optimum cruise level and this will change as fuel is burnt and the aircraft becomes lighter. In simple terms, it’s more economical to fly as close to optimum as possible, i.e. as high as possible. The factors that will influence this are wind (you wouldn’t climb 2,000ft into significantly higher headwinds), congestion along airways etc.
Having said all of that, the speeds that are actually flown often differ from those calculated by the cost index. Within terminal areas, flow of traffic will usually mean that ATC mandate your speed in the descent. Similarly, cruise speed will often be modified to manage traffic along a busy airway.
Regarding climb rates (someone else mentioned), 99% of the time an aircraft will simply climb at the highest rate it can achieve at climb thrust. No commercial airliners climb at such high rates to impact passenger comfort. At normal weights, you’d expect to reach a cruise level of 37,000ft in about 25 minutes with the highest climb rates at the lowest levels and slower rates of climb as the air becomes thinner at altitude. A 747 won’t be climbing at anything like 10,000fpm unless doing a very brief zoom climb.